Question

The time between arrivals of small aircraft at a county airport is exponentially distributed with a mean of one hour. Round the answers to 3 decimal places (e.g. 98.765). What is the probability that more than three aircraft arrive within an hour? If 30 separate one-hour intervals are chosen, what Is the probability that no interval contains more than three arrivals? Determine the length of an interval of time (In hours) such that the probability that no arrivals occur during the interval is 0.14.

Answer 1

Answer:

a) 0.019

b) 0.563

c) x = 1.966 hours

Step-by-step explanation:

E(X) = 1

Exponential random variable's probability function is given as

P(X=x) = λ e^(-λ.x)

The cumulative distribution function is given as

P(X ≤ x) = 1 - e^(-λ.x)

a) The time between the arrivals of small aircraft at a county airport that is exponentially distributed.

But the number of planes that land every hour will be obtained using the Poisson distribution formula.

It is the best for discrete systems.

Poisson distribution formula is given as

P(X = x) = (e^-λ)(λˣ)/x!

λ = 1 aircraft per hour.

The probability that more than three aircraft arrive within an hour = P(X > 3)

P(X > 3) = 1 - P(X ≤ 3) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3)]

P(X > 3) = 1 - 0.98101 = 0.01899 = 0.019 to 3 d.p

b) If 30 separate one-hour intervals are chosen, what Is the probability that no interval contains more than three arrivals

Probability of one 1-hour interval not containing more than 3 arrivals = 1 - P(X > 3)

= 1 - 0.01899 = 0.98101

Probability that thirty 1-hour intervals will not contain more than 3 arrivals = (0.98101)³⁰ = 0.5626 = 0.563 to 3 d.p

c) Determine the length of an interval of time (In hours) such that the probability that no arrivals occur during the interval is 0.14

We can now use the cumulative distribution function for exponential random variable for this

P(X ≤ x) = 1 - e^(-λ.x)

P(X > x) = 1 - P(X ≤ x)

P(X > x) = e^(-λ.x)

λ = 1, x = ?,

0.14 = e⁻ˣ

e⁻ˣ = 0.14

In e⁻ˣ = In 0.14 = -1.966

-x = -1.966

x = 1.966 hours

Hope this Helps!!!